Help to win at Bingo

In a recent book written by Joseph Granville, he says that you can increase your chances of winning at Bingo by 50%. The idea is to choose your Bingo cards so that the numbers on the cards do not have unlikely sequences on them. Examples:

Card A   Card B B I N G O B I N G O 2 24 36 51 63 8 29 34 56 75 3 18 39 49 64 5 24 31 46 69 6 22 XX 50 66 1 23 XX 60 68 7 16 31 47 61 12 16 45 59 62 5 20 35 60 65 15 17 42 54 61

According to the book, Card A has what the book calls "bad symmetry" with clustered numbers and Card B has "excellent symmetry" with a random distribution of numbers like it should be in bingo. This all sounds reasonable but is complete foolishness, mathematically. I believe that every card has the same exact chance of winning as any other card. The following three hypotheses will prove my point unequivocally.

Proof #1 : To simplify the situation, let's start with smaller B(ingo) cards:

Card X   Card Y B B 1 12 2 5 3 7

We will choose numbers between 1 and 15, and 3 in a row wins. According to the book, card X has "bad symmetry," while card Y has "good symmetry." Which B(ingo) card is more likely to win then? Mathematics says that every number is equally likely. 1 is as likely as 7. In fact the odds are 1/15 that any given number will be chosen on the first pick. Well, then it must be the later picks which make X a bad card. For the purposes of this proof, let's assume that we are tied with one hit each (3 & 12) after 4 picks.

Card X   Card Y B B 1 X 2 5 X 7

The book might now argue that 2 or 1 are now not very likely. True. But then neither are 5 or 7. No combination of two specific numbers is very likely. In fact the chance of hitting a 1 (or a 5 or any other number) is now 1/11. I can argue that in all future situations the actual numbers on the cards do not matter. This proof is valid.

Proof #2 : Now let's play real bingo. You can have card B, while I will choose a made-up card C, chosen so that it has no numbers in common with card B.

Card C   Card B B I N G O B I N G O 2 18 32 47 63 8 29 34 56 75 3 19 33 48 64 5 24 31 46 69 4 20 XX 49 65 1 23 XX 60 68 6 21 35 50 66 12 16 45 59 62 7 22 36 51 67 15 17 42 54 61

But now, we are going to disguise all of the numbers in our Bingo game. A 2 becomes an 8, a 3 becomes a 5, etc., based on this table:

1 » 4 16 » 21 31 » 33 46 » 48 61 » 67 2 » 8 17 » 22 32 » 34 47 » 56 62 » 66 3 » 5 18 » 29 33 » 31 48 » 46 63 » 75 4 » 1 19 » 24 34 » 32 49 » 60 64 » 69 5 » 3 20 » 23 35 » 45 50 » 59 65 » 68 6 »12 21 » 16 36 » 42 51 » 54 66 » 62 7 »15 22 » 17 37 » 37 52 » 52 67 » 61 8 » 2 23 » 20 38 » 38 53 » 53 68 » 65 9 » 9 24 » 19 39 » 39 54 » 51 69 » 64 10 »10 25 » 25 40 » 40 55 » 55 70 » 70 11 »11 26 » 26 41 » 41 56 » 47 71 » 71 12 » 6 27 » 27 42 » 36 57 » 57 72 » 72 13 »13 28 » 28 43 » 43 58 » 58 73 » 73 14 »14 29 » 18 44 » 44 59 » 50 74 » 74 15 » 7 30 » 30 45 » 35 60 » 49 75 » 63

We also use the same table to disguise all of our numbered balls which the machine will choose. Here are our disguised cards:

Card C   Card B B I N G O B I N G O 8 29 34 56 75 2 18 32 47 63 5 24 31 46 69 3 19 33 48 64 1 23 XX 60 68 4 20 XX 49 65 12 16 45 59 62 6 21 35 50 66 15 17 42 54 61 7 22 36 51 67

As you can see, our disguised card B' looks just like our old card C and C' looks like B. But they are not the old cards. Under our fake numbers are the old numbers and we are just using a code for each number. Well, by the definition of "symmetry" in the book, we find that C' now has "good symmetry" and B' now has "bad symmetry." And now, C' is much more likely to win than B'. That is a contradiction. The book says that each card is both a better bet, and a worse bet.

Proof #3 : Here is my final proof. I played 10,000 games on my computer using only cards A and B to see if one scores much better than the other. After 10,000 games, card A won 4911 to 4865, with 224 ties; pretty even. Statistics shows that this data supports my hypothesis that the two cards have equal chances to win. And it does not support the alternative hypothesis that B is better than A. This test definitely refutes any 50% improvement in the winning chances as the author claims.